# Automorphism commutators by Miller G.A.

By Miller G.A.

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**Example text**

The surfaces fall into two categories depending on whether the infinitesimal linear transformation leave invariant or permute their points at infinity. However, we can neglect the first category, for it is clear from the above that it includes only developable surfaces or more specifically only cylindrical surfaces. Thus, we can assume that our infinitesimal transformation permutes the points at infinity among themselves. Then, the plane at infinity will undergo projective transformation and will contain ∞1 curves which remain invariant.

Thus, every two-dimensional integral manifold z1 = f, z2 = ϕ of our involutory system F1 = 0, F2 = 0, F3 = 0 contains ∞2 different characteristic strips and only ∞1 different characteristic curves. Every such curve is the locus of ∞1 characteristic strips. 46. Thus, the semilinear partial differential equation Ω = 0 in a fourdimensional space has ∞5 characteristic strips, but only ∞4 characteristic curves, among which ∞1 always pass through a point x, y, z1 , z2 , in general position. It follows from the above that every point x, y, z1 , z2 is the vertex of an elementary cone containing only ∞1 directions of motion and thus being defined not by one, but by two Monge equations Φ1 (x, y, z1 , z2 , dx1 , dy1 , dz1 , dz2 ) = 0, Φ2 = 0.

51. Note that the form of equations (c) is determined not only by the form of the original equations Fj = 0. Taking different equations z1 = 30 Sophus Lie ϕ1 (x, a), . . , zm = ϕm (x, a) as a basis, one obtains different systems (c). However, it is possible to determine a system of first-order partial differential equations with independent variables x1 , . . , xn , z1 , . . , zm and with one unknown function V satisfied by all manifolds z1 = ϕ1 , . . , zm = ϕm . Indeed, the equations F j x1 , .